Computational analysis of electrode structure and configuration for efficient and localized neural stimulation

Abstract

Neuromodulation technique using electric stimulation is widely applied in neural prosthesis, therapy, and neuroscience research. Various stimulation techniques have been developed to enhance stimulation efficiency and to precisely target the specific area of the brain which involves optimizing the geometry and the configuration of the electrode, stimulation pulse type and shapes, and electrode materials. Although the effects of electrode shape, size, and configuration on the performance of neural stimulation have individually been characterized, to date, there is no integrative investigation of how this factor affects neural stimulation. In this study, we computationally modeled the various types of electrodes with varying shapes, sizes, and configurations and simulated the electric field to calculate the activation function. The electrode geometry is then integratively assessed in terms of stimulation efficiency and stimulation focality. We found that stimulation efficiency is enhanced by making the electrode sharper and smaller. A center-to-vertex distance exceeding 100 µm shows enhanced stimulation efficiency in the bipolar configuration. Additionally, the separation distance of less than 1 mm between the reference and stimulation electrodes exhibits higher stimulation efficiency compared to the monopolar configuration. The region of neurons to be stimulated can also be modified. We found that sharper electrodes can locally activate the neuron. In most cases, except for the rectangular electrode shape with a center-to-vertex distance smaller than 100 µm, the bipolar electrode configuration can locally stimulate neurons as opposed to the monopolar configuration. These findings shed light on the optimal selection of neural electrodes depending on the target applications.

Introduction

Neuromodulation is a technique that modulates neuronal activities by applying external energy such as electrical, magnetic, optical, and mechanical energy to neurons [1–3]. Among the various energy modalities used to modulate neural activity, electrical neuromodulation is considered a gold standard method as it is the most effective method to stimulate the neuron and it is considered a safe method during stimulation [1]. Moreover, the electrical stimulation device can be miniaturized so that it can be portably attached to the body or implanted inside [4, 5]. Due to these fascinating properties, electrical neuromodulation is widely employed in neural prosthetic devices, therapy, and neuroscience research [4, 6, 7].

Depending on the applications, the configuration and structure of the electrode have been determined. For instance, a linear array of round-shaped stimulation electrodes was used in a cochlear implant which is one of the most successful neural prosthetic devices [8]. For this case, a monopolar electrode configuration was employed to stimulate the auditory ganglion cell. When electrical stimulation is used in neuroscience research, selective stimulation of neurons is essential for unraveling the function of neural circuitries. In this case, the microelectrode or patch clamp technique is used [10].

Although many types of neural electrodes with varying shapes, sizes, and configurations have been developed, little work has been performed to increase the stimulation efficiency and to control the area of neural stimulation. Attempts to enhance neural stimulation efficiency by modifying the electrode shape have been investigated as the efficient stimulation may reduce power consumption and thereby minimize the device size [11–15]. Golestanirad et al. proposed fractal electrodes for efficient neural stimulation. By increasing the edginess of the electrode surface the neural activation was enhanced [11]. Grill et al. hypothesized that a high-perimeter electrode enhances the neural stimulation efficiency as the variation of current density increases thereby having a larger activation function [12]. By computational modeling, the DBS electrode having a serpentine perimeter was able to efficiently stimulate the neuron by reducing the power consumption up to ~ 20%.

The effect of the electrode configuration on the selective activation of neurons was investigated [13, 14, 16]. Gomez-Tames et al. computationally found that direct subcortical stimulation using a bipolar electrode produces more selective activation while a monopolar electrode was more robust and effective [13]. Spencer et al. demonstrated that combining electrodes from six hexagonally located electrodes as well as varying the charge injected through each selected, electrode could steer the target location [16].

Although much effort has been put into determining the optimum electrode structural configuration, there is no integrative and quantitative analysis of how electrode design and configuration influence efficient and selective neural stimulation. Moreover, the electrode size, shape, and configuration collectively affect the performance of neural stimulation and hence it may not be suitable to conclude by considering individual factors alone. For instance, it has been conventionally known that bipolar configuration can locally activate the neuron which aligns with the finding of Gomez-Tames et al. However, the monopolar configuration can be more effective in the selective stimulation of neurons compared to the bipolar configuration [17] inferring that other factors may have influenced the performance of the neural stimulation.

In this paper, we computationally analyze the effect of electrode shape, size, and configuration integratively on neural stimulation. As we target extracellular stimulation, the activating function is employed to quantitatively analyze the degree of neural activation. The effect of neural stimulation was investigated in terms of efficient stimulation to reduce power consumption and selective stimulation to minimize unwanted stimulation of off-target neurons.

Methods

Extracellular neural stimulation

When applying current/voltage to the neuron via an extracellularly placed electrode, the electric potential vicinity to the neuron changes. When looking into the electric potential along the axon, the fluctuation of the electric potential modulates neural activities. At the valley of the electric potential fluctuation, the current is injected into the neuron and thereby increasing the membrane potential. The location where the valley of extracellular electric potential is determined by the positive peak of the second derivatives of electric potential along the axon fiber as shown in Eq. (1)

$$f\left(x,,,t\right)=\frac{{\partial }^2{V}_e(x,t)}{\partial x^2}$$ 1

where, $x$ is the length coordinate of the axon fiber, $t$ is the time, $V_e$ is the extracellular electric potential [18]. The $f\left(x,,,t\right)$ is called an activating function which reflects the activation of axon fiber upon extracellular stimulation [18].

The second derivative of the electric potential ($f_x$ and $f_y$) along $x$ and $y$-directions reflects the activation of neurons which lie along $x$ and $y$-directions, respectively. We determined the activation function of neurons lying in all directions by the root-mean-square of the first derivative of the electric field ($E_x$ and $E_y$) along $x$ and $y$-directions in 2-D model (Eq. 2).

$$AF=\sqrt{{\left(f_x\right)}^2+{\left(f_y\right)}^2}=\sqrt{{\left(-,\frac{\partial E_x}{\partial x}\right)}^2+{\left(-,\frac{\partial E_y}{\partial y}\right)}^2}$$ 2

The magnitude of the activation function (AF) is used to determine the efficiency of neural stimulation. The focality of stimulation is determined by the area of the activation function over specific limit. We assume that half of the maximum point of the activation function is the threshold point of neural activation. The full area at the half maximum (FAHM) of the activation function ($FAHM\left(A,F\right)$) is defined as the sum of the area that exceeds the half maximum point of the activation function. We also evaluate the $D(E_{\mathit{max}}/2)$ defined as the depth that reaches the half maximum point of the electric field in 3-D bipolar electrode model to assess the stimulation depth according to distance between the electrode pair.

Modeling of neural electrode

Neural electrodes were modeled with various shapes, sizes, and configurations (Table 1). At first, the shape of the electrode is modeled as a regular polygon with four different numbers of edges. For each shape, electrodes with four different sizes were designed. The electrode size was characterized by its center-to-vertex distance of the regular polygon which ranges from 50 to 1000 $\mu m$. Two different electrode configurations, a monopolar and a bipolar electrode configuration were designed. For the bipolar electrode configuration, a reference electrode is designed to have dimensions identical to the stimulation electrode, with a 1 mm separation between them. We also changed the distance between the stimulation electrode and the reference electrode in the range of 0.25–2 mm for the bipolar electrode configuration whose center-to-vertex distance of the stimulation electrode is fixed as 50 $\mu m$. In the monopolar configuration, the reference electrode was designated as the boundary of the gray matter, while the stimulation electrode was positioned at the center of the brain tissue. The effects of electrode parameters on neural stimulation efficiency and the area of neural activations were investigated.

Table 1.

Parameters and their corresponding values of the stimulation neural electrode

Parameters	Values
Number of edges of a regular polygon	4, 6, 8, and 10
Center-to-vertex distance ($\mu m$)	50, 100, 500, and 1000
Electrode configuration	Monopolar and bipolar
Distance between stimulation and reference electrode (mm)	0.25, 0.5, 1, and 2

COMSOL simulation environment

A finite element method (FEM) simulation was performed to calculate the spatial distribution of the activation function arising from the stimulation neural electrode. We built a 2-D model of the neural electrode using a COMSOL (COMSOL Inc., Burlington, MA, USA). The gold metal neural electrode was modeled which was placed inside the gray matter (electrical conductivity of gold: $4.11\times {10}^7$ S/m, electrical conductivity of gray matter: 0.27 S/m). As cathodic stimulation is more preferable in activating neural response, -1 V was assigned at the boundary of the electrode for cathodal neural stimulation [18]. The potential of the reference electrode was set as 0 V for the bipolar electrode configuration. The size the boundary of the simulation was determined by changing its size while monitoring the AF. The square-shaped monopolar configuration with a center-to-vertex distance of 50 $\mu m$ is used. The boundary size was determined based on the point at which the AF no longer changes with changes in the boundary size (Fig. 1). Here, we fixed the size of the simulation boundary as 10 mm.

Fig. 1.

Maximum activation function as a function of size of the simulation boundary. The monopolar electrode configuration with − 1 V input is applied to the stimulation electrode

After 2D modeling, we discretized the model using a fine mesh in the COMSOL. An electric current solver is used to solve all the electric potential, the electric field, and the current in the neural tissue. Briefly, the electric current solver solves the scalar electric potential ($V$) and the electric field ($\mathit{E}$) using the Eq. (3).

$$\mathit{E}=-\mathrm{\nabla }V$$ 3

The current density ($\mathit{J}$) was then calculated using the simple Ohm’s law using Eq. (4).

$$\mathit{J}=\sigma \mathit{E}$$ 4

where, $\sigma$ is the electric conductivity.

For a 2-D model, a triangular mesh is refined with custom element size, where the maximum and minimum element size was set to 12 μm and 1 μm, respectively. In the 3-D model, designed when calculating the $D(E_{\mathit{max}}/2)$, a tetrahedral mesh is refined with custom element size and the maximum and minimum element size was set to 50 μm and 1 μm, respectively. The maximum element growth rate of 2 and curvature factor of 0.2 were employed and the resolution of narrow region was assigned as 1. After solving the electric field, piecewise linear interpolation was employed. The electric field and AF are extracted from the COMSOL and calculation including the interpolation were conducted in the Python.

Result

Effect of electrode shape on neural activation

The effect of the electrode shape on neural activation was investigated. At first, the monopolar electrode configuration with the center-to-vertex distance of 50 μm was employed. We found that the $\mathit{AF}$ was localized near the vertex (Fig. 2). The maximum value of the $\mathit{AF}$, noted as $AF\left(m,a,x\right)$, decreases as the number of edge of the regular polygon increases (Fig. 3a) and becomes $1.5\times {10}^8\mathrm{V}/{\text{m}}^2$ when using decagon-shaped stimulation electrode. Meanwhile, the $FAHM\left(A,F\right)$ increases as the number of edges of the regular polygon increase and reaches $2.20\times {10}^{-9}{\text{m}}^2$ at decagon-shaped electrode (Fig. 3b). Moreover, we confirmed that the $AF\left(m,a,x\right)$ is linearly proportional to the magnitude of the input voltage (Fig. 4) due to the linearity of Maxwell’s equation [19]. This indicates that the efficiency of neural stimulation or $AF\left(m,a,x\right)$ can be modified by changing the input voltage. In contrast to the $AF\left(m,a,x\right)$, the $FAHM\left(A,F\right)$ does not change as applied input voltage changes further confirming that distribution of $FAHM\left(A,F\right)$ is linearly proportional to the input voltage. For instance, the $FAHM\left(A,F\right)$ values for electrodes with edge numbers of 4, 6, 8, and 10 are shown as $3.27\times {10}^{-10}$, $1.27\times {10}^{-9}$, $2.18\times {10}^{-9}$, and $2.20\times {10}^{-9}{m}^2$, respectively, regardless of the applied input voltage in all cases (− 0.1 V, − 1 V, and − 10 V).

Fig. 2.

Distribution of the activation function of a square, b hexagon, c octagon, and d decagon. The monopolar electrode configuration with − 1 V input is applied to the stimulation electrode and the center-to-vertex distance is set to 50 $\mu m$. Scale bar = $20\mu m$. The color bar indicating the value of the $\mathit{AF}$ applies to all figures

Fig. 3.

The effects of number of edges of a regular polygon for the stimulation electrode on the a maximum activation function ($AF\left(m,a,x\right)$) and b full area at the half maximum of the activation function ($FAHM\left(A,F\right)$)

Fig. 4.

Effects on the maximum activation function while changing the applied voltage to the stimulation electrode. Lines with circles, crosses, and squares represent − 0.1 V, − 1 V, and − 10 V, respectively. The monopolar electrode configuration with the center-to-vertex distance of 50 $\mu m$ is used

Effect of electrode size on neural activation

To find the effect of the electrode size on neural activation, the regular hexagon with the monopolar electrode configuration is employed. As the size of the electrode increases, the $AF\left(m,a,x\right)$ decreases logarithmically and becomes $7.61\times {10}^7[\text{V}/{\text{m}}^2]$ when the center-to-vertex distance becomes 1 mm (Fig. 5a). The effect of electrode size was also investigated for other electrode shapes and found that the profile of $AF\left(m,a,x\right)$ over electrode size conforms to that of the regular hexagon while their values are scaled by factor governed by the shape as noted in the Sect. 3.1.

Fig. 5.

The effects of electrode size and shape on the maximum activation function ($AF(max)$). The line with circles, crosses, squares, and asterisks represent square, hexagon, octagon, and decagon, respectively

Effect of electrode configuration on neural activation

The effect of electrode configuration on neural activation was investigated. At first, the distribution of AF was visualized for the bipolar electrode configuration (Fig. 6). Analogous to the monopolar configuration, $\mathit{AF}$ was localized near the vertex especially near the stimulation electrode and the greater $AF\left(m,a,x\right)$ was monitored when the number of edges gets smaller. We next compared the bipolar and monopolar electrode configurations on the effects of the neural activation. To compare the region of the neural activation, the difference of the $FAHM\left(A,F\right)$ ($FAHM\left(A,F\right)\left(B,i,p,o,l,a,r\right)-FAHM\left(A,F\right)\left(M,o,n,p,o,l,a,r\right)$) measured at each electrode configuration were obtained for various electrode sizes and shapes. We found that the bipolar electrode configuration has a smaller $FAHM\left(A,F\right)$ indicating that it has higher focality (Fig. 7a). The discrepancy maximizes as the size of the stimulation electrode gets bigger with a smoother shape which is the decagon-shaped electrode with the center-to-vertex distance of 500 μm. Interestingly, monopolar configuration showed smaller $FAHM\left(A,F\right)$ only for the rectangular electrode with the center-to-vertex distance of 50 μm. As the stimulation electrode in the bipolar configuration is separated from the reference electrode by 1 mm, we investigated how this separation affects the stimulation focality. As the stimulation electrode gets closer to the reference electrode, the bipolar electrode performs more focal stimulation compared to the monopolar electrode except for the rectangular shaped electrode (Fig. 7b).

Fig. 6.

Distribution of the activation function of a and a′ square, b and b′ hexagon, c and b′ octagon, and d and a′ decagon. The stimulation electrode is located on the left and the reference electrode is located on the right and the distributions of the activation function are visualized for both electrodes (a, b, c, and d). Zoom-in images of the stimulation electrode (a′, b′, c′, and d′). Scale bars for (a, b, c, and d) and (a′, b′, c′, and d′) are 20 $\mu m$ and 200 $\mu m$, respectively. The color bar at the right side of (d) and (d′) apply to (a, b, and c) and (a′, b′, and c′), respectively

Fig. 7.

Effect of electrode configuration on neural activation. Difference of bipolar and monopolar configuration in terms of a, b the full area at the half maximum of the activation function ($FAHM\left(A,F\right)\left(B,i,p,o,l,a,r\right)-FAHM\left(A,F\right)(Monopolar)$) and c, d the maximum activation function ($AF\left(m,a,x\right)\left(B,i,p,o,l,a,r\right)-AF\left(m,a,x\right)(Monopolar)$)

We also compared the stimulation efficiency by calculating the difference of the $AF\left(m,a,x\right)$ obtained at two different electrode configurations (($AF\left(m,a,x\right)\left(B,i,p,o,l,a,r\right)-AF\left(m,a,x\right)\left(M,o,n,p,o,l,a,r\right)$). When the center-to-vertex distance of the electrode exceeds 100 μm, the bipolar configuration exhibits higher $AF\left(m,a,x\right)$ compared to the monopolar configuration (Fig. 7c). Conversely, the monopolar configuration has higher $AF\left(m,a,x\right)$ when the center-to-vertex distance of the electrode is smaller than 100 μm. For the electrode with the center-to-vertex distance of 100 μm, the bipolar configuration showed higher stimulation efficiency for octagon and decagon-shaped electrodes. We then investigated the effects of the separation between the reference and the stimulation electrodes after fixing the size of the electrode with the center-vertex-distance of 50 μm. The monopolar electrode configuration has greater $AF\left(m,a,x\right)$ than the bipolar configuration when the separation distance is larger than 0.5 mm (Fig. 7d). As the separation distance gets smaller and the shape of the boundary becomes sharper, the bipolar configuration has a bigger $AF\left(m,a,x\right)$

Subsequently, the effects of the separation distance between the stimulation and the reference electrodes on the electric field’s extent, particularly in the depth direction, was investigated. Rectangular-shaped electrode with a center-to-vertex distance of 50 um is designed and the electric field and $D(E_{\mathit{max}}/2)$ at the midpoints of two electrodes are measured. As the two electrodes become closer, $D(E_{\mathit{max}}/2)$ becomes smaller and its electric field becomes larger (Fig. 8). These results indicate that smaller separation distance is beneficial for achieving more localized stimulation along the depth.

Fig. 8.

Effects of the separation distance between the stimulation and the reference electrode on the extent of the electric field measured along the depth direction. The electric field is measured at the midpoints of two electrodes. a The depth where the electric field becomes half of its maximum point and b its electric field is plotted as the separation distance

Discussion

In this study, we explored how the design, and the arrangement of electrodes impact the activation of neurons, particularly in terms of stimulation efficiency and stimulation focality. Given that stimulation efficiency has implications for tissue safety and the power usage of electrical stimulation devices, and stimulation focality influences the number of neurons activated, it is crucial to optimize electrode designs that consider both efficiency and focality simultaneously. This study significantly influences the electrode design for a wide range of applications in electrical neuromodulation (Fig. 9).

Fig. 9.

Distribution of the activation function of circular shaped electrode. a Monopolar and b and c bipolar electrode configuration is used. − 1 V input is applied to the stimulation electrode and the center-to-vertex distance is set to 50 $\mu m$. Scale bars for (a, b) and (c) are 20 $\mu m$ and 200 $\mu m$, respectively. The color bar indicating the value of the $\mathit{AF}$ applies to all figures

To enhance the stimulation efficiency by increasing the $AF\left(m,a,x\right)$, the number of edges of the regular polygon should be decreased as well as the size of electrode should be reduced. The bipolar electrode configuration in combination with the large electrode (the center-to-vertex distance exceeding 100 μm) and the short distance (< 1 mm) between the reference and the stimulation electrode is beneficial in increasing the stimulation efficiency. When using the monopolar electrode configuration, the small electrode (the center-to-vertex distance less than 100 μm) is advantageous in improving the neural stimulation efficiency.

We further elucidate the effect of a number of edges in the regular polygon on the activation function. Since the $\mathit{AF}$ is localized in the vertex, the location in the boundary where the edge is bent is crucial. As the electric field is localized at the tip of the electrode and the bumpy circumference of the electrode creates strong fluctuation of the electric field which both increases the activation function which is the second-order derivative of the electric field in terms of distance. This indicates that the electrode with a sharper edge exhibiting a more pronounced edge effect is beneficial to enhance the stimulation efficiency.

There are several benefits of enhancing the stimulation efficiency. First, high stimulation efficiency could reduce the power consumption during neural stimulation. As many neuromodulation devices are portable and many of them are implantable devices, they are powered by batteries that require frequent recharging or replacement [20, 21]. For instance, DBS device requires additional surgery to replace the battery [20]. The other reason the improving stimulation efficiency is to improve the safety during the neural stimulation. Voltage is applied to the electrode during the neural stimulation which may electrolysis the water and/or degrade the electrode in reaction with the chemical component in the neural tissue [22]. Improving the stimulation efficiency reduces the applied voltage at the electrode and hence reduces the possible damage during neural stimulation.

With regards to the selective neural activation, the localized stimulation is obtained by reducing the number of edges of the regular polygon. Bipolar configuration stimulates neurons more locally compared to the monopolar configuration except for the rectangular-shaped electrode. For the monopolar configuration, the electrode size smaller than 100 μm with a rectangular-shaped electrode is good for localized stimulation. These findings indicate that bipolar configuration does not always guarantee localized stimulation but rather optimization of electrode configuration including its geometry is essential.

As the number of the vertex increases and reaches infinity then the shape of the electrode becomes circular shape. In this case, the distribution of $\mathit{AF}$ becomes smooth along the outer boundary of the electrode (Fig. 7a). The trend of electrode shape when the center to vertex distance of 50 μm on the neural activation still applies to the circular-shaped electrode showing smaller $AF\left(m,a,x\right)$ while the bigger $FAHM\left(A,F\right)$ compared to those of decagon. The distribution of $\mathit{AF}$ for the bipolar electrode shows similar trends with the monopolar electrode.

As the AF distribution of the circular electrode is uniform, it can modulate neurons more uniformly at the expense of stimulation efficiency. Therefore, it is essential to optimize both the shape and size of the electrode to achieve a balance between increased stimulation efficiency and uniform neural stimulation for specific applications. For instance, to modulate a small population of neurons, a small-sized electrode with a sharper vertex polygon is preferable to maximize neural stimulation efficiency. On the other hand, when aiming for uniform modulation of a large population of neurons, the circular electrode is more effective, even though it sacrifices some stimulation efficiency.

Meshing parameters are crucial in FEM simulation as the size of the mesh significantly impacts simulation accuracy and time. Larger mesh sizes can decrease accuracy but reduce simulation time, while smaller mesh sizes enhance accuracy at the cost of increased simulation time. Selecting an appropriate range for mesh size, defined by minimum and maximum values, is essential. Moreover, it is important to determine the ‘resolution of narrow region’ which controls how fine the mesh created in the narrow region. When this resolution varies with the electrode size, the size and the number of mesh especially at the electrode boundary varies which results in varying computational resolution. Notably, we observed that the full area at half maximum (FAHM) of the activation function is affected by mesh size, particularly near the boundary.

To date various neuromodulation schemes have been developed to enhance the stimulation efficiency and selective activation of the neurons. It has been known that cathodal stimulation is preferable in stimulating axon with lower threshold and with more localized region compared to anodal stimulation [23]. Duration of the stimulation current affect the performance of the neural stimulation. The pulse duration that matches the chronaxie time can stimulate the neuron efficiently [24]. Current injected at different location with varying onset time [25], frequency [26], amplitude [27] can localized the stimulation current at the target location. The waveform shape such as rising exponential and the pulse shape also affects the stimulation efficiency [28]. These factors all together can be incorporated with the electrode shape, size, and configuration to enhance the performance of the neural stimulation.

Conclusion

In conclusion, we have investigated various electrode design factor for efficient and localized neural stimulation. Our result suggests that the efficient neural stimulation can be obtained by designing the sharp electrode and minimizing the electrode size. The effect of electrode configuration on the stimulation efficiency varies with the size of electrodes. Bipolar electrode with small electrode (the center-to-vertex distance less than 100 μm) and a short separation (< 1 mm) distance between the reference and the stimulation electrode can enhance the stimulation efficiency. For monopolar electrode configuration, the design of the electrode in the reverse way can increase the stimulation efficiency. We also found that localized stimulation can be achieved by designing the sharp electrode. The electrode configuration has varying degrees of influence on the area of neuron activated. The monopolar electrode with a rectangular shape and small electrode (the center-to-vertex distance less than 100 $\mu m$) can stimulate more local areas of the neural population. For other cases, the bipolar configuration can locally stimulate the neurons compared to that of the monopolar configuration. Our computational study of electrode design to maximize the stimulation efficiency and local stimulation of the neuron can be a design criterion of electrodes for various applications of electrical neuromodulation in clinics.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1C1C1010505, RS-2023-00217893), and by BK21PLUS, Creative Human Resource Education and Research Programs for ICT Convergence in the 4th Industrial Revolution.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Consent to participate

Not applicable.

Consent to publish

Not applicable.